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The Intensional Side of Algebraic-Topological Representation Theorems

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The Intensional Side of Algebraic-Topological Representation Theorems The intensional side of algebraic-topological representation theorems Sara Negri Department of Philosophy, History, Culture and Art Studies, P.O. Box 24, FIN-00014, University of Helsinki Finland [email protected] Abstract Stone representation theorems are a central ingredient in the metathe- ory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems. 1 Introduction Algebraic-topological representation theorems have a long history, dating back to the work of Marshall Stone on the representation of Boolean algebras (1936), with the spaces that arise from such representation bearing his name. As em- phasized in the historical introduction of Johnstone (1982), the legacy of Stone representation theorems has a↵ected almost all fields of mathematics, having been a tool not only in logic but also in topology, with the study of its fun- damental structures from a lattice-theoretic and categorical viewpoint (locale theory and topos theory), and encompassing also the areas of algebraic geometry and functional analysis. The common aspect of such theorems is the representation of algebraic struc- tures in terms of set-theoretic ones, often endowed with a topology. Ordered 1 algebraic structures are an abstraction of the properties of Boolean operations of intersection, union, and complementation in sets, with the order corresponding to set inclusion, and can be considered as an intensional way of conceptualizing the extensional properties of topological spaces. The shift from intensional to extensional definitions is a delicate one, and in fact, the move requires the use of a non-constructive principle to assert the existence of ideal elements. The same shift is encountered when topological notions are formalised intensionally in approaches such as formal or pointfree topology1, specifically in proving that these new topologies can be made equivalent to the traditional, point-set ones. The wide area of algebraic logic has its foundations in duality theory that in turn depends largely on algebraic-topological representations. In particular, Stone representation theorems are a central ingredient in the metatheory of philosophical logics. In this field, one of the most well known uses of Stone- type representations is in the embedding of intuitionistic logic into modal logic, achieved, through completeness, with the representation of complete Heyting algebras as closure spaces.2 The representation of Heyting algebras in terms of closure algebras is used in— and in a way corresponds to—the proof of faithfulness of the translation of intuitionistic logic, Int, into the modal logic S4. In turn, the embedding of intuitionistic logic into the modal logic S4 was obtained as an attempt to give a provability interpretation of intuitionistic logic. G¨odel’s (1933) sound transla- tion from Int into S4 was proved faithful as long as 15 years later by McKinsey and Tarski (1948). The result was then generalised by Dummett and Lemmon (1959) who extended it to intermediate logics between Int and classical logic and the corresponding intemerdiate logics between S4 and S5. Like the result by McKinsey and Tarski, Dummett and Lemmon’s proof used the same indirect semantic method, namely the completeness of intuitionistic logic with respect to Heyting algebras (there called Brouwerian algebras) and of S4 with respect to topological Boolean algebras (called closure algebras). The algebraic part of the proof consists in the representation of Heyting algebras as the opens of topolog- ical Boolean algebras. The proof is therefore indirect because it is obtained by algebraic means through completeness, and non-constructive because of the use of Stone representation of distributive lattices, in particular Zorn’s lemma. In more recent literature, surveyed by Wolter and Zakharyaschev (2014), embed- ding results are established semantically and by use non-constructive principles. All these results obtained by semantic means establish correspondences with re- spect to derivability between superintuitionistic systems and normal extensions of S4. In particular, they do not show how to translate a proof from one system to the other. The interest in obtaining a direct proof of the faithfulness result was therefore motivated not only by the need of adhering to a strict purity of 1Cf. Sambin (1987). 2Heyting algebras are also called pseudo-Boolean algebras,e.g.inRasiowaandSikorski (1963). 2 methods. It has been shown (Dyckho↵ and Negri 2012) that logical embeddings can be obtained as translations of derivations, in a direct way that does not employ the non-constructive principles typical of the representation theorems. The method is based on the uniform development of analytic proof systems for the logics in question. Analytic proof systems stem from the tradition of Gentzen systems of sequent calculi, with the characteristic feature that in a proof only syntactic objects ranging in a set which is predetermined by the statement to be proved are used. If one uses axiomatic systems for embedding results, the soundness part of the proof can be given, as G¨odel did, by induction on derivations. For the faithfulness part, however, this is not in general possible. The reason is that when proceeding inductively one needs to have in the derivations premisses to which the inductive hypothesis can be applied, but the inductive hypothesis can be applied only to formulas in the range of the translation. Without a calculus with sufficient analyticity properties, in particular with an axiomatic system, the premisses of the last rule used in the derivation can be very far from a formula of that form, and the inductive hypothesis cannot in general be applied. Sequent calculi with cut elimination and the subformula property are a remedy to this shortcoming of axiomatic systems and thus give the potentially appropriate tool for establishing results of this kind directly. When such systems can be developed in a uniform way for a family of log- ics, through the definition of a common core calculus to which further rules are added, they can be used not just for studying the relation of logical consequence of a given logic, but for establishing metatheoretic results, such as embeddings, on the relationship between di↵erent logics. Analyticity and uniformity are the key features that reduce the proof of faithfulness of those embeddings to an induction on derivations, and thus one avoids the need of the detour through completeness and Stone-representations. Analyticity gives full control on the structure of derivations in the frame classes considered. Uniformity, on the other hand, guarantees that the same procedure works in all the corresponding systems because it operates only on the common core calculi and leaves un- changed the added rules. As a further evidence of its generality and flexibility, we show how the method can be employed to establish also the embedding of subintuitionistic logics into the corresponding modal logics below S4. The general picture that emerges with the development of uniform analytic methods for non-classical logics not only shows that the indirect and non- constructive proofs based on Stone representation can be avoided in the proof of such embeddings, but also suggests a change in perspective, that is, to priv- ilege the use of logical methods, and more specifically, proof-theoretic meth- ods, to recover the intensional, pointfree part of representation results. By the identification of free Heyting algebras with the Lindenbaum algebras of intu- itionistic logic, the logical embedding gives an embedding of freely generated Heyting algebras in closure algebras. Finally, thanks to the precise sense in which the inductive generation of syntactic objects in a sequent calculus paral- lels the inductive generation of equalities between terms in a free algebra, the 3 proof-theoretic embedding can then be interpreted as a reduction of classes of word problems. The paper is organized as follows: Section 2 gives background notions and a summary of the current literature and results on the algebraic-topological representation theorems and their use in logical embeddings. Some features concerning modal embedding results obtained by traditional (semantic) means are analyzed: in particular, the proof of such results uses indirect and non- constructive proofs based on Stone-type representation theorems. Section 3 presents two main lines of enquiry directed at the isolation and avoidance of non-constructive principles in Stone-like theorems and their applications. The first is given by the development of constructive mathematics within formal topology, discussed in Section 3.1, the second by analytic proof theory, espe- cially labelled sequent calculi. Section 3.2 shows how uniform analytic methods for non-classical logic can be employed to establish a proof-theoretic embedding of intermediate and subintuitionistic logic into the corresponding modal logic. Finally, Section 4 addresses
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